Tailing off dates from the mid-19th century. As you’d expect, the origin of the verb “to tail” (in all its various senses) is the same as the noun, deriving from the Old English tægel, which in turn derives from older Germanic roots and possibly from the Proto-Indo-European roots before that. Off was an Old English word with a similar set of meanings as it has now, and again came to us from Germanic and possibly PIE roots.

Hitchhike (note that the form without a space is more conventional) dates to the 1920s. Hitch comes from the 13th century verb hytchen meaning to move suddenly, and beyond that, orig obs alas. Hike, meaning walk, appears as hyke in the early 19th century, and not much else can be said of its origins with certainty.

Youth hostel dates from the 1930s, and this phrase has been formed in a fairly straightforward fashion from the pre-existing words youth and hostel. Youth is from the Old English geoguth and derives from older Germanic roots. Hostel first appeared in English in the 13th century, apparently adopted straight from a French word meaning the same thing, which in turn derived from Latin roots meaning host.

There is no English phrase Elder hostel but there is an organisation Elderhostel, which has been running since the 1970s. The name of the organisation was formed from the words elder and hostel. The latter I’ve already covered. Elder existed in Old English in exactly the same form and meaning as it has now! It derived from older Germanic roots and there is some suggestion of a connection to PIE roots.